Solving Code Formation Using Permutations

The question asks for the number of distinct codes formed by selecting 5 symbols from a set of 8 different symbols, where the order of symbols matters and no symbol is repeated.

This scenario describes a permutation, as we are arranging a subset of items where the order is significant and repetition is not allowed.

Permutation Formula

The number of permutations of choosing '$k$' items from a set of '$n$' distinct items (order matters, no repetition) is calculated using the formula:

$P(n, k) = \frac{n!}{(n-k)!}$

Applying to the Problem

Here, we have:

• Total number of symbols ($n$) = 8
• Number of symbols to select for the code ($k$) = 5

Substitute these values into the permutation formula:

$P(8, 5) = \frac{8!}{(8-5)!} = \frac{8!}{3!}$

Calculation Steps

Expand the factorial terms:

$P(8, 5) = \frac{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{3 \times 2 \times 1}$

Simplify by canceling the '$3!$' ($3 \times 2 \times 1$) from the numerator and denominator:

$P(8, 5) = 8 \times 7 \times 6 \times 5 \times 4$

Perform the multiplication:

$P(8, 5) = 56 \times 6 \times 5 \times 4$

$P(8, 5) = 336 \times 5 \times 4$

$P(8, 5) = 1680 \times 4$

$P(8, 5) = 6720$

Result

Thus, there are 6720 different codes that can be formed.